___________________________________________________________________________ Operations Research  Jan Fábry PERT Project Management

___________________________________________________________________________ Operations Research  Jan Fábry Project Management PERT  Probabilistic time – 3 estimates Optimistic Pessimistic Most likely Expected completion time

a ij m ij b ij OptimisticPessimistic Most Likely 0 Time ___________________________________________________________________________ Operations Research  Jan Fábry Project Management  Activity time β-distribution μ ij ExpectedPERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management  Expected completion time  Standard deviation PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management 1)Using CPM ( t ij = μ ij ) Critical path 2)Expected duration of the project Variance of the project duration PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Duration of the project - normal distribution T M M + σ M - σ T – real duration M – expected duration – standard deviation σ – standard deviation PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management 3)Probability analysis desired time T D probabilty of completion time within T D desired time T D p PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management 3)Probability analysis desired probability p zpzp completion time PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. Activity Description a ij m ij b ij μ ij Predecessors A Songs Selection None BMastering7898A C PM Elaborating 5676A D Customers Analysis 5797A E PM Production C, D F PM to PH 34115E G Customers Selection 2343D PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. Activity Description a ij m ij b ij μ ij Predecessors H Make CD Copies B, D I Data to PH 2343G J Laser Print F, I KMailing68108 H, J PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc A B D C D1D1 E FH D2D2 GI J KPERT Critical path:

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. Activity Description a ij m ij b ij μ ij σ ij A Songs Selection /6 D Customers Analysis 57974/6 E PM Production /6 F PM to PH /6 J Laser Print /6 KMailing681084/6 PERT Critical activities

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. M = 48 Expected duration of the project  = 3 Std. deviation of the project duration  2 = 9 Variance of the project duration PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Time (days) Duration of the project Example - Music DM, Inc. PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. Probability analysis Desired time T D = 45 PERT

___________________________________________________________________________ Operations Research  Jan Fábry Project Management Example - Music DM, Inc. Probability analysis Desired probability = 0.95 z p = PERT

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Inventory  Stored for use in future  Examples of inventories - raw material - finished goods - semi-finished products - spare parts

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Reasons for Maintaining Inventories  Fluctuating demand  Delayed supply  Savings on ordering cost  Benefits on large quantities  Inflation

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Management  How much to order?  When to order?  Objective – minimize total cost

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Partial Inventory Cost  Ordering & setup cost – fixed charge cost frequency of ordering  Unit purchasing cost – variable cost quantity discounts

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Partial Inventory Cost  Holding & carrying cost – variable cost - storage cost (space) - store keeping operations (movement) - insurance & taxes - interest (invested capital ) - oportunity cost - spoilage or obscolescence

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Partial Inventory Cost  Shortage (stockout) cost – variable cost - idled production (idled machine) - placing special expensive expediting order - loss of customers due to late deliveries

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Inventory Terminology  Inventory level  Demand & depletion Rate of demand  Depletion rate Available size of the inventory

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Inventory Terminology  Reordering Reorder point = signal inventory level Time Placing order Lead time Receiving shipment & Replenishment

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Inventory Terminology  Shortage Time Lead time Inventory level = 0 Optimal reorder date Placing order Receiving shipment Shortage !!

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Inventory Terminology  Safety stock Buffer being built to prevent shortage In case of probabilistic demand

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models  Deterministic models Rate of demand is known with certainty Probabilistic demand (random variable)  Probabilistic models

StaticDynamicNonstationaryStationary ___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Demand Classification DeterministicProbabilistic Demand Rate of demand - known with certainty - constant Rate of demand - known with certainty - not constant Probability density function – unchanhged over time Probability density function – unchanhged over time Probability density function – varies in time Probability density function – varies in time Increasing Level of Mathematical Difficulty

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Deterministic Models  Basic economic order quantity model (EOQ model)  EOQ model with back orders allowed (with planned shortages)  Economic production lot size model  EOQ model with quantity discounts

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Models  Probabilistic model with continuous demand  Single-period decision model